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On Chevalley restriction theorem

Eugene Tevelev
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Abstract

Let $g$ be a complex semisimple Lie algebra with adjoint group $G$. Suppose that $σ$ is an involutive automorphism of $g$. Then $σ$ induces uniquely an involution of $G$ also denoted by $σ$, let $K=G^σ$ be a subgroup of $σ$-fixed points. Consider a direct decomposition $g=k+p$ of $g$ into eigenspaces for $σ$. Then $p$ is a $K$-module. Denote by $a\subset p$ any maximal abelian ad-diagonalizable subalgebra. Consider the ``baby Weyl group'' $W=N_K(a)/Z_K(a)$. Let $ψ: C[p]^K\to C[a]^W$ be a restriction map of algebras of invariants. Then the famous Chevalley restriction theorem states that $ψ$ is an isomorphism. The aim of this paper is prove the following Theorem. The restriction map $ψ: C[p\times p]^K\to C[a\times a]^W$ is surjective.

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Eugene Tevelev. 1999-01-15. On Chevalley restriction theorem. https://arxiv.org/abs/math/9901060

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